Intitulé |
Numerical Algebraic Geometry and Algebraic Numerical Analysis |

Organisateurs |
Michele Bolognesi et Daniele A. Di Pietro |

Objectifs |
Ce groupe de travail a pour but d'explorer des développements récents au carrefour entre la géométrie algébrique et l'analyse numérique. Au passage, il permettra de renforcer les liens entre les équipes de mathématiques fondamentales et appliquées de l'IMAG. |

Structure |
Deux-trois séances par an animées par des orateurs internationaux de très haut niveau |

Acknowledgements |
ANR projects HHOMM and fast4hho
EDF |

**Spline functions for geometric modeling and numerical simulation**

In geometric modeling, a standard representation of shapes is based on parameterized surfaces or volumes, which involve piecewise polynomial functions also known as splines. We will first describe these splines, in the univariate case, how they are characterized and what are their main properties. We will present some extensions to higher dimensions and how they are used for modeling shapes, in several domains of applications. We will also illustrate several exemples of spline constructions, that have been investigated. In the last decades, a new paradigm called isogeometric analysis has emerged, which uses splines functions not only to describe the geometry but also to approximate functions on this geometry. We will briefly describe how this approach is working in numerical simulation, what are its main characteristics and we will provide some illustrations. Analyzing the spaces of spline functions associated to a given domain partition or a given abstract topology is an important problem for geometric modeling and numerical simulations. We will present some of the know results and some problems still open. We will describe algebraic-geometric techniques, which provide a better insight on these spline spaces, in particular for the analysis of their dimension and for the construction of basis. This analysis involves topological complexes and homological tools, which will be explained by examples. Spline spaces over triangular meshes or T-meshes, in dimension 2 and 3, as well as extensions to geometrically smooth spline spaces (if time permits) will be considered in details.

**From the finite element method to the finite element exterior calculus**

The finite element exterior calculus (FEEC) has been developed in the early 2000s as a mathematical framework that uses the calculus of differential forms for the formulation of the finite element method. The aim of this lecture is to revisit some of the main aspects of FEEC. We shall use as a main motivation the approximation of Laplace equation in mixed form and of the time harmonic formulation of Maxwell's equations. It is remarkable that, besides providing a more elegant and comprehensive understanding of the existing theory, the FEEC allows for the development of new finite elements and for the solution of problems that would be difficult, if not impossible, to address with more traditional tools

Updated 14/10/2019